K-Theory and Characteristic Classes in Topology and Complex Geometry (A Tribute to Atiyah and Hirzebruch)

K-theory and characteristic classes in topology and complex geometry (a tribute to Atiyah and Hirzebruch)

Claire Voisin

CNRS, Institut de math´ematiquesde Jussieu

CMSA Harvard, May 25th, 2021 Plan of the talk

I. The early days of Riemann-Roch • Characteristic classes of complex vector bundles • Hirzebruch-Riemann-Roch. Ref. F. Hirzebruch. Topological Methods in Algebraic Geometry (German, 1956, English 1966) II. K-theory and cycle class • The Atiyah-Hirzebruch spectral sequence and cycle class with integral coeﬃcients. • Resolutions and Chern classes of coherent sheaves Ref. M. Atiyah, F. Hirzebruch. Analytic cycles on complex manifolds (1962) III. Later developments on the cycle class • Complex cobordism ring. Kernel and cokernel of the cycle class map. • Algebraic K-theory and the Bloch-Ogus spectral sequence The Riemann-Roch formula for curves

• X= compact Riemann surface (= smooth projective complex curve). E → X a holomorphic vector bundle on X. •E the sheaf of holomorphic sections of E. Sheaf cohomology H0(X, E)= global sections, H1(X, E) (eg. computed as Cˇech cohomology). Def. (holomorphic Euler-Poincar´echaracteristic) χ(X,E) := h0(X, E) − h1(X, E). • E has a rank r and a degree deg E = deg (det E) := e(det E). • X has a genus related to the topological Euler-Poincar´echaracteristic: 2 − 2g = χtop(X).

• Hopf formula: 2g − 2 = deg KX , where KX is the canonical bundle (dual of the tangent bundle). Thm. (Riemann-Roch formula) χ(X,E) = deg E + r(1 − g) Sketch of proof Sketch of proof. (a) Reduction to line bundles: any E has a ﬁltration by subbundles Ei such that Ei/Ei+1 is a line bundle. The 3 quantities r, χ and deg are additive under short exact sequences.

(b) Reduction to OX : L= holomorphic line bundle on X, x ∈ X. Line bundle L(−x) whose sheaf of sections is L ⊗ Ix, with short exact sequence 0 → L ⊗ Ix → L → Cx → 0. One has deg L(−x) = deg (L) − 1, χ(X,L(−x)) = χ(X,L) − 1. ⇒ (*) χ(X,L) = χ(X, OX ) + deg L.

(c) Serre duality ⇒ χ(X,KX ) = −χ(X, OX ). Formula (*) for KX then gives 2χ(X, OX ) = −deg KX hence χ(X, OX ) = 1 − g. qed • Surfaces. For a holomorphic line bundle L on a projective surface X, one “easily” gets using the Riemann-Roch formula on curves, 2 L −KX ·L (∗∗) χ(X,L) = χ(X, OX ) + 2 . • Serre duality gives χ(X, OX ) = χ(X,KX ); already contained in (**). • Hirzebruch uses the Hodge index theorem + topological formulae for the 2 c1(X) +c2(X) signature ⇒ Noether formula χ(X, OX ) = 12 . Sketch of proof Sketch of proof. (a) Reduction to line bundles: any E has a ﬁltration by subbundles Ei such that Ei/Ei+1 is a line bundle. The 3 quantities r, χ and deg are additive under short exact sequences.

(c) Serre duality ⇒ χ(X,KX ) = −χ(X, OX ). Formula (*) for KX then gives 2χ(X, OX ) = −deg KX hence χ(X, OX ) = 1 − g. qed • Surfaces. For a holomorphic line bundle L on a projective surface X, one “easily” gets using the Riemann-Roch formula on curves, 2 L −KX ·L (∗∗) χ(X,L) = χ(X, OX ) + 2 . • Serre duality gives χ(X, OX ) = χ(X,KX ); already contained in (**). • Hirzebruch uses the Hodge index theorem + topological formulae for the 2 c1(X) +c2(X) signature ⇒ Noether formula χ(X, OX ) = 12 . Chern classes of complex vector bundles (Chern, Borel, Borel-Hirzebruch) • Chern. E= complex diﬀerentiable vector bundle on a manifold X. 1 C-linear Hermitian connection ∇ on E curvature R∇ = 2iπ ∇ ◦ ∇ and k real closed forms Tr R∇ of degree 2k real cohomology classes.

• Chern classes ck(E) :=“k-th symmetric functions of the eigenvalues of R∇”. Related to the classes above by the Newton formulas. 2 • L=complex line bundle on X first Chern class c1(L) ∈ H (X, Z). 1 0 ∗ 2 Defined using the map H (X, (C ) ) → H (X, Z) induced by the 0 0 ∗ exponential exact sequence 0 → Z → C → (C ) → 1. Thm. (Axiomatic construction/characterization of Chern classes) There 2i exist unique Chern classes ci(E) ∈ H (X, Z) for any E,X, with total P Chern class c(E) = i ci(E) satisfying the following axioms. (i) Contravariant functoriality. (ii) (Whitney formula) c(E ⊕ F ) = c(E) · c(F ). (iii) c(L) = 1 + c1(L), where c1(L) is as defined above. The proof uses the splitting principle : Given E → X, there exists a ∗ ∗ ∗ ∗ f : Y → X such that f : H (X, Z) → H (Y, Z) is injective and f E is a direct sum of line bundles. Chern classes of complex vector bundles (Chern, Borel, Borel-Hirzebruch) • Chern. E= complex differentiable vector bundle on a manifold X. 1 C-linear Hermitian connection ∇ on E curvature R∇ = 2iπ ∇ ◦ ∇ and k real closed forms Tr R∇ of degree 2k real cohomology classes.

• Chern classes ck(E) :=“k-th symmetric functions of the eigenvalues of R∇”. Related to the classes above by the Newton formulas. 2 • L=complex line bundle on X first Chern class c1(L) ∈ H (X, Z). 1 0 ∗ 2 Defined using the map H (X, (C ) ) → H (X, Z) induced by the 0 0 ∗ exponential exact sequence 0 → Z → C → (C ) → 1. Thm. (Axiomatic construction/characterization of Chern classes) There 2i exist unique Chern classes ci(E) ∈ H (X, Z) for any E,X, with total P Chern class c(E) = i ci(E) satisfying the following axioms. (i) Contravariant functoriality. (ii) (Whitney formula) c(E ⊕ F ) = c(E) · c(F ). (iii) c(L) = 1 + c1(L), where c1(L) is as defined above. The proof uses the splitting principle : Given E → X, there exists a ∗ ∗ ∗ ∗ f : Y → X such that f : H (X, Z) → H (Y, Z) is injective and f E is a direct sum of line bundles. Chern character and Todd genus; formalism of virtual roots

• Virtual roots and symmetric functions. For any symmetric polynomial f in k variables λ1, . . . , λk, one has a polynomial Pf in the symmetric functions σi of λ1, . . . , λk, such that Pf (σ·) = f(λ·).

• Works as well with formal series. If f has coeﬃcients in A, so does Pf . ∗ • E a vector bundle of rank k on X with Chern classes ci(E) ∈ H (X, Q). ∗ For any f as above Pf (c·(E)) ∈ H (X, Q). The λi implicitly used in the function f are called the virtual roots of the Chern polynomial. When the vector bundle is a direct sum of line bundles, one can take λi = c1(Li).

• In general, the λi can be realized as cohomology classes only on a splitting manifold Y → X for E. P • Chern character: ch E = i exp λi. Obviously ch (E ⊕ F ) = ch E + ch F , ch (E ⊗ F ) = ch E · ch F . • Todd genus. td E = Q λi . Obviously i 1−exp(−λi) td (E ⊕ F ) = td E · td F . Hirzebruch-Riemann-Roch formula

• E=complex vector bundle on X= complex manifold. TX has a complex structure Chern classes ci(E), cj(TX ). • Holomorphic structure on E sheaf E of holomorphic sections, cohomology groups Hi(X, E) and holomorphic Euler-Poincar´e P i i characteristic χ(X,E) := χ(X, E) = i(−1) h (X, E) (X compact). Thm. (Hirzebruch-Riemann-Roch formula) One has R χ(X,E) = X ch E · td X =: T0(X,E).

• The χy-genus. TX is a holomorphic vector bundle, hence also ∗ P p p ΩX = TX . Deﬁne χy(E) := p y χ(X,E ⊗ ΩX ).

• Obvious. χ(X,E) = χ0(E).

• Less obvious, due to Serre. For the trivial bundle OX , one has χ−1(X, OX ) = χtop(X). n Proof. Holomorphic de Rham complex 0 → OX → ΩX → ... → ΩX → 0. This is a resolution of the constant sheaf C. qed

• Ty-genus Ty(X,E) : plug-in y in the formal expression for ch E · td X, P eg chy(E) = i exp(1 + y)λi. Hirzebruch-Riemann-Roch formula

• The χy-genus. TX is a holomorphic vector bundle, hence also ∗ P p p ΩX = TX . Deﬁne χy(E) := p y χ(X,E ⊗ ΩX ).

• Obvious. χ(X,E) = χ0(E).

• Ty-genus Ty(X,E) : plug-in y in the formal expression for ch E · td X, P eg chy(E) = i exp(1 + y)λi. Strategy of the proof in the projective case • Reduction to the line bundle case. Work on P(E) and the Hopf line bundle H on P(E). Leray spectral sequence ⇒ χ(P(E),H) = χ(X,E). • Reduction to the absolute case (trivial line bundle). If D ⊂ X is a smooth hypersurface, and L = OX (−D) = ID, one has 0 → L → OX → OD → 0 so χ(X,L) = χ(X, OX ) − χ(D, OD). Use also 0 → L|D → ΩX|D → ΩD → 0. • Absolute case. Index τ(X) for X real oriented of dimension 2n: τ(X) = 0 if n is odd, otherwise τ(X) := signature of intersection pairing n on H (X, R). Thom cobordism ⇒ τ(X)= polynomial in the Pontryagin classes of X. If X is almost complex: get Chern number of X. Hirzebruch: this is T1(X). Thm. (Hodge index thm) If X is a complex projective manifold, one has P p τ(X) = p χ(X, ΩX ) =: χ1(X, OX ).(True for X complex compact).

• ⇒ equality χ1(X, OX ) = T1(X).

• Functional equation for χy-genus and Ty-genus + equality for y = 1 ⇒ χ0(X, OX ) = T0(X) for X a split manifold, and ﬁnally for any X. qed Strategy of the proof in the projective case • Reduction to the line bundle case. Work on P(E) and the Hopf line bundle H on P(E). Leray spectral sequence ⇒ χ(P(E),H) = χ(X,E). • Reduction to the absolute case (trivial line bundle). If D ⊂ X is a smooth hypersurface, and L = OX (−D) = ID, one has 0 → L → OX → OD → 0 so χ(X,L) = χ(X, OX ) − χ(D, OD). Use also 0 → L|D → ΩX|D → ΩD → 0. • Absolute case. Index τ(X) for X real oriented of dimension 2n: τ(X) = 0 if n is odd, otherwise τ(X) := signature of intersection pairing n on H (X, R). Thom cobordism ⇒ τ(X)= polynomial in the Pontryagin classes of X. If X is almost complex: get Chern number of X. Hirzebruch: this is T1(X). Thm. (Hodge index thm) If X is a complex projective manifold, one has P p τ(X) = p χ(X, ΩX ) =: χ1(X, OX ).(True for X complex compact).

• ⇒ equality χ1(X, OX ) = T1(X).

• Functional equation for χy-genus and Ty-genus + equality for y = 1 ⇒ χ0(X, OX ) = T0(X) for X a split manifold, and ﬁnally for any X. qed Complex K-theory

• For a topological space X, K0(X) is the abelian group with generators the isomorphism classes [E] of complex vector bundles E on X, and 0 relations [E ⊕ F ] = [E] + [F ]. For pointed space (X, x), K = rank 0 at x. 0 • Holomorphic variant. X= complex manifold. Kan(X) is the abelian group with generators the isomorphism classes [E] of holomorphic vector bundles E on X, and relations [G] = [E] + [F ] whenever there exists an exact sequence 0 → E → G → F → 0 of holomorphic vector bundles. • Due to the Whitney axiom, Chern classes factor through K0. The Chern character gives a ring homomorphism to rational cohomology. • Atiyah-Hirzebruch introduce K∗: 1 0 1 0 K (X) := Ker (K (X × S ) → K (X)) + Bott periodicity. For a pair 0 (X,Y ) (say of CW -complexes), let K0(X,Y ) := K (X/Y ). Long exact sequence (*) K−1(Y ) → K0(X,Y ) → K0(X) → K0(Y ) → K1(X,Y ) → ... The Atiyah-Hirzebruch spectral sequence

• X a CW-complex. Xi ⊂ X is the i-skeleton of X, union of cells of dimension ≤ i. • One gets a decreasing ﬁltration of the cochain complex by subcomplexes ∗ p p,q C (X,X ) and a spectral sequence with E1 = 0 for q 6= 0, p,0 p p p−1 p,0 p,0 p E1 = C (X /X ), E2 = E∞ = H (X, Z). • Using (*), Atiyah and Hirzebruch construct a similar spectral sequence for K-theory. pq p+q pq Thm. There exists a spectral sequence E2 ⇒ K (X) with E2 = 0 if pq p q is odd, E2 = H (X, Z) if q is even.

• (Formal). The differential dr vanishes for even r. • With Q-coefficients, the differentials must vanish (compare with cohomology). pq pq Cor. One has E2 = E∞ if odd (i) H (X, Z) = 0 or ∗ (ii) H (X, Z) has no torsion. Resolutions and characteristic classes of coherent sheaves • Let Z ⊂ X be a closed analytic subset in a complex manifold. Coherent sheaves IZ ⊂ OX , OZ = OX /IZ . • In the smooth projective case: any coherent sheaf admits a (finite) locally free resolution. Follows from (a) local statement, (b) any coherent sheaf H admits a surjective quotient map F → H → 0 with F locally free. • Not true in the general compact complex case: Thm. (Voisin 2002) Take X = T very general complex torus of dimension 3, x ∈ T a point. Then Ix does not admit a locally free resolution. • X complex compact. Atiyah-Hirzebruch use locally free resolutions of coherent sheaves by real analytic complex vector bundles: 0 → Fn → ... Fi ... → F0 → Hω → 0. Thus any coherent sheaf H has a 0 Q i i class in K (X). One has c(H) = i c(Fi) , i = (−1) . Thm. (Atiyah-Hirzebruch, Grothendieck-Riemann-Roch) Z ⊂ X closed 0 analytic of codimension k. OZ has a class in K (X,X \ Z) and k−1 2k (*) ck(OZ ) = (−1) (k − 1)![Z] in H (X, Z). 2k • Here [Z] ∈ H (X, Z) is the cycle class of Z. Resolutions and characteristic classes of coherent sheaves • Let Z ⊂ X be a closed analytic subset in a complex manifold. Coherent sheaves IZ ⊂ OX , OZ = OX /IZ . • In the smooth projective case: any coherent sheaf admits a (finite) locally free resolution. Follows from (a) local statement, (b) any coherent sheaf H admits a surjective quotient map F → H → 0 with F locally free. • Not true in the general compact complex case: Thm. (Voisin 2002) Take X = T very general complex torus of dimension 3, x ∈ T a point. Then Ix does not admit a locally free resolution. • X complex compact. Atiyah-Hirzebruch use locally free resolutions of coherent sheaves by real analytic complex vector bundles: 0 → Fn → ... Fi ... → F0 → Hω → 0. Thus any coherent sheaf H has a 0 Q i i class in K (X). One has c(H) = i c(Fi) , i = (−1) . Thm. (Atiyah-Hirzebruch, Grothendieck-Riemann-Roch) Z ⊂ X closed 0 analytic of codimension k. OZ has a class in K (X,X \ Z) and k−1 2k (*) ck(OZ ) = (−1) (k − 1)![Z] in H (X, Z). 2k • Here [Z] ∈ H (X, Z) is the cycle class of Z. Atiyah-Hirzebruch counterexamples to the integral Hodge conjecture

Conj. (Hodge conjecture) Let X=projective complex manifold and 2k P α ∈ H (X, Q) be of Hodge type (k, k). Then α = i αi[Zi], with αi ∈ Q, Zi ⊂ X closed of codim. k. Rem. Equivalent formulations, using resolutions and formula (*):

α ∈ hck(F)iQ, F= coherent sheaf on X, or α ∈ hck(F)iQ, F= locally free coherent sheaf on X. • The three statements are wrong in the compact Kählercase (Voisin). • Z-coefficients. Wrong (Atiyah-Hirzebruch). Thm. (Atiyah-Hirzebruch) Let X= compact complex manifold, Z ⊂ X 2k closed analytic subset of codim k with class [Z] ∈ H (X, Z). Then [Z] is annihilated by all the differentials dr, r ≥ 3 of the A-H spectral sequence. • The example. 2-torsion cohomology class α of degree 4 on a smooth projective manifold X which is not annihilated by d3 ⇒ not a cycle class. • Construction: Serre’s trick : finite group G acting on projective space N N CP complete intersection X ⊂ CP on which G acts freely X/G. Atiyah-Hirzebruch counterexamples to the integral Hodge conjecture

• Milnor construction of MU∗(pt). 1) Generators: compact diﬀerentiable manifolds M of dim ∗ + a virtual complex structure on TM . 2) Complex cobordism relations: N = diﬀerentiable manifold with boundary ∂N and virtual complex structure on TN , hence virtual complex structure on T∂N .

• Complex cobordism group MU∗(X),X=manifold: 1) Generators: compact diff. manifolds M of dim. ∗, + diff. map f : M → X + virtual ∗ complex structure on virtual normal bundle Nf := f TX − TM . 2) Complex cobordism relations: N = differentiable manifold with boundary ∂N, F : N → X differentiable map and virtual complex

structure on NF , hence virtual complex structure on NF|∂M .

• Map o : MU∗(X) ⊗MU∗(pt) Z → H∗(X, Z), (M, f) → f∗[M]. Iso. ⊗Q. Thm. (Totaro) X compact complex manifold, Z ⊂ X closed analytic subset of codim k. Then (1) [Z] ∈ Im o. (2) ∃ canonical lift of [] to refined cycle class []˜. (1) Follows from Hironaka resolution of singularities. Allows to reinterpret the Atiyah-Hirzebruch obstruction by computing in MU∗. Nontopological obstructions to the algebraicity of integral Hodge classes • The Atiyah-Hirzebruch-Totaro obstruction for an integral cohomology class α on X =complex compact manifold to be algebraic is topological. ⇒ The class α does not become a cycle class on a deformation of X. n • Kollár’sexamples. Let X ⊂ P , n ≥ 4, be a smooth hypersurface of 2n−4 degree d. Lefschetz thm on hyperplane sections ⇒ H (X, Z) = Zα, with deg α = 1. If X contains a line ∆, α = [∆]. Thm. (Kollár) If X is very general of degree pn−1, p ≥ n − 1 prime, any curve C ⊂ X has degree divisible by p. Hence α is not algebraic. • The proof is by specialization of X to the image of a generic map n−1 n P → P given by polynomials of degree p. 2 • n = 4. X as above, S = surface with 0 6= β ∈ H (S, Z) of p-torsion. ∗ ∗ 6 p-torsion class γ = pr1α ^ pr2β ∈ H (Y, Z),Y = X × S. Thm. (Soulé-Voisin) The p-torsion class γ is not algebraic on Y for X very general. (The prime p is arbitrarily large, the dimension is fixed.) • The torsion class γ is algebraic on Y for special X, eg X containing a line. Nontopological obstructions to the algebraicity of integral Hodge classes • The Atiyah-Hirzebruch-Totaro obstruction for an integral cohomology class α on X =complex compact manifold to be algebraic is topological. ⇒ The class α does not become a cycle class on a deformation of X. n • Kollár’sexamples. Let X ⊂ P , n ≥ 4, be a smooth hypersurface of 2n−4 degree d. Lefschetz thm on hyperplane sections ⇒ H (X, Z) = Zα, with deg α = 1. If X contains a line ∆, α = [∆]. Thm. (Kollár) If X is very general of degree pn−1, p ≥ n − 1 prime, any curve C ⊂ X has degree divisible by p. Hence α is not algebraic. • The proof is by specialization of X to the image of a generic map n−1 n P → P given by polynomials of degree p. 2 • n = 4. X as above, S = surface with 0 6= β ∈ H (S, Z) of p-torsion. ∗ ∗ 6 p-torsion class γ = pr1α ^ pr2β ∈ H (Y, Z),Y = X × S. Thm. (Soulé-Voisin) The p-torsion class γ is not algebraic on Y for X very general. (The prime p is arbitrarily large, the dimension is fixed.) • The torsion class γ is algebraic on Y for special X, eg X containing a line. The Bloch-Ogus spectral sequence

• Let X be a complex algebraic manifold. X(C) has two topologies, the Euclidean and Zariski topologies continuous map f : Xan → XZar. • The Bloch-Ogus spectral sequence is the Leray spectral sequence of f. i i • A abelian group. H (A) := R f∗A, sheaf associated to presheaf i U 7→ H (Uan,A) on XZar. p,q p q p+q • E2 = H (XZar, H (A)) ⇒ H (Xan,A). p,q Thm. (Bloch-Ogus) (a) One has E2 = 0 for p > q. k,k k (b) A = Z. E2 is isomorphic to Z (X)/alg. k,k k,k 2k (c) The induced map E2 → E∞ ,→ H (X, Z) is the cycle class map k 2k []: Z (X)/alg → H (X, Z). k P • Group of cycles Z (X) = { i niZi, codim Zi = k}. Def. X projective. Z,Z0 ⊂ X are algebraically equivalent if ∃ smooth projective curve C, a cycle Z in C × X (ﬂat over C) and two points t, t0 0 of C such that Zt − Zt0 = Z − Z as cycles of X. The Bloch-Ogus spectral sequence

• Let Griffk(X) := Ker [ ] ⊂ Zk(X)/alg. Analyzing the Bloch-Ogus spectral sequence in degree 4, get: Cor. (Bloch-Ogus) (k = 2) Exact sequence 3 0 3 d2 2 H (X, Z) → H (XZar, H (Z)) → Griff (X) → 0. • Thm. (Griffiths) There exist smooth projective threefolds X with 2 Griff (X) ⊗ Q 6= 0. 0 3 3 Cor. (Bloch-Ogus) The group H (XZar, H (Q))/H (X, Q) can be nonzero. k k • Define H (C(X),A) := lim H (U, A). → ∅6=U⊂X, Zar. open Thm. (Bloch-Ogus) The space H0(X, Hk(A)) identifies with k res k−1 Ker (H (C(X),A) → ⊕D divisorH (C(D),A)). • Whether a class with no residues comes from a class on the total space had been asked by Atiyah and Hodge (Integrals of the second kind on an algebraic variety. Ann. of Math. 1955) (and established in degree ≤ 2). The Bloch-Ogus spectral sequence, contd